Concept of functions: domain, range, graphs

IB Mathematics Applications and Interpretation HL· 25 min read

1. Definition of a Function★☆☆☆☆⏱ 10 min

📘 Definition

Function

f(x)

A relation that maps every input in the set of valid inputs to exactly one output. Multiple inputs can map to the same output, but one input can never map to multiple outputs.

Example:

f(x) = 3x - 2 $i s a f u n c t i o n,$ x = y^2 $i s n o t a f u n c t i o n f or a l l r e a l$ x$.

To quickly check if a graph represents a function, we use the vertical line test: if any vertical line crosses the graph more than once, the relation is not a function.

📐 Worked Example

Determine if $x^{2} + y^{2} = 9$ (a circle of radius 3 centered at the origin) is a function using the vertical line test.

1
Rearrange the equation to solve for $y$ :
2
$y^2 = 9 - x^2 \implies y = \pm\sqrt{9 - x^2}$
3
For any $x$ between $- 3$ and $3$ , there are two distinct values of $y$ for the same input $x$ .
4
When we draw any vertical line in this interval, it crosses the circle twice, which violates the function definition.
5
Conclusion: The circle is not a function.

Exam tip:

Only one-to-one functions pass the horizontal line test, but all functions pass the vertical line test. Don't confuse the two in exams!

2. Domain and Range★★☆☆☆⏱ 15 min

📘 Definition

Domain and Range

Domain: All valid input $x$ values where the function is defined. Range: All possible output $f (x)$ values the function produces.

Example:

Domain of $f (x) = \frac{1}{x}$ is all real numbers except $0$ ; range is also all real numbers except $0$ .

The most common domain restrictions come from three rules: (1) you cannot divide by zero, (2) you cannot take the square root of a negative number, (3) you cannot take the logarithm of a non-positive number. Always check for these first when finding domain.

📐 Worked Example

Find the domain and range of $f (x) = 6 - 2 x$ . State your answer in interval notation.

1
First find domain: the expression under the square root must be non-negative:
2
$6 - 2x \geq 0 \implies 2x \leq 6 \implies x \leq 3$
3
Write domain in interval notation, including the endpoint because $x = 3$ gives $0 = 0$ , which is valid:
4
Domain = $(- \infty, 3]$
5
Find range: the principal square root function only outputs non-negative values. As $x$ decreases, $6 - 2 x$ increases from 0 to infinity:
6
Range = $[0, \infty)$

3. Key Features of Function Graphs★★☆☆☆⏱ 12 min

The graph of a function $f (x)$ is the set of all points $(x, f (x))$ on the coordinate plane. For exams, you will be expected to identify the following key features from an equation or graph: x-intercepts (roots, where $f (x) = 0$ ), y-intercept (where $x = 0$ ), turning points, horizontal/vertical asymptotes, and intervals of increase/decrease.

📐 Worked Example

For $f (x) = (x + 1) (x - 4)$ , find the x-intercepts, y-intercept, and state the range of the function.

1
Find x-intercepts where $f (x) = 0$ :
2
$(x+1)(x-4) = 0 \implies x = -1, x = 4$
3
X-intercepts are at $(- 1, 0)$ and $(4, 0)$ .
4
Find y-intercept where $x = 0$ :
5
$f(0) = (0+1)(0-4) = -4$
6
Y-intercept is at $(0, - 4)$ .
7
This is a quadratic with positive leading coefficient, so it opens upwards with minimum value at the vertex. The vertex is at $x = 1.5$ , $f (1.5) = - 6.25$ , so the range is:
8
$[-6.25, \infty)$

4. Common Pitfalls

Wrong move:

Writing the domain of $x - 4$ as $(4, \infty)$ instead of $[4, \infty)$ .

Why:

Forgetting that $x = 4$ gives an output of 0, which is perfectly valid.

Correct move:

Always include endpoints when the restriction inequality is non-strict ( $\geq$ or $\leq$ ).

Wrong move:

Claiming a relation is not a function because a horizontal line crosses it twice.

Why:

Confusing the vertical line test (for functions) with the horizontal line test (for one-to-one functions).

Correct move:

Only the vertical line test determines if a relation is a function, regardless of what horizontal lines do.

Wrong move:

Including the $x$ value that makes a denominator zero in the domain of a rational function.

Why:

Forgetting that division by zero is undefined, so the function does not exist at that point.

Correct move:

Always exclude any $x$ that makes any denominator equal to zero, regardless of other factors.

Wrong move:

Stating the range of $f (x) = x^{2} + 2$ as $[2, 10]$ when the domain is given as $0 \leq x \leq 3$ .

Why:

Forgetting that the maximum value of a quadratic with positive leading coefficient on a closed interval occurs at one of the endpoints.

Correct move:

Check the value of the function at both endpoints of the domain interval when finding the range of a restricted quadratic.

Wrong move:

Writing the range of $f (x) = x - 4$ as $(- \infty, \infty)$ , matching the form of the domain.

Why:

Confusing domain and range, and forgetting the principal square root only produces non-negative outputs.

Correct move:

Always calculate domain and range separately, don't assume they are the same.

5. Quick Reference Cheatsheet

Function Type	Common Domain Restriction	Typical Range
Linear $f (x) = m x + c$	None	$(- \infty, \infty)$
Quadratic $f (x) = a x^{2} + b x + c, a > 0$	None	$[min (f), \infty)$
Rational $f (x) = 1/ g (x)$	Exclude $x : g (x) = 0$	Depends on $g (x)$
Square Root $f (x) = g (x)$	Require $g (x) \geq 0$	$[0, \infty)$
Logarithm $f (x) = lo g (g (x))$	Require $g (x) > 0$	$(- \infty, \infty)$

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2025 · 1
Find domain of composite rational function
2024 · 2
State range of quadratic function
2023 · 1
Use vertical line test to identify functions

Going deeper

syllabusIB AI HL Syllabus: Topic 2 Functions

What's Next

The concepts you learned here are the foundation for all function work in IB AI HL, appearing in every topic from calculus to statistical modelling. Mastering domain, range, and function definitions early will help you avoid avoidable mistakes in every exam question involving functions. Next, you will build on this knowledge to explore composite and inverse functions, then learn about specific function types used for modelling, before applying these concepts to differentiation and integration. Solid understanding here is critical for success in higher level topics.